1 Since the estimates in this publication are based on information obtained from occupants of a sample of dwellings, they are subject to sampling variability. That is, they may differ from those estimates that would have been produced if all dwellings had been included in the survey. One measure of the likely difference is given by the standard error (SE), which indicates the extent to which an estimate might have varied by chance because only a sample of dwellings was included. There are about two chances in three (67%) that a sample estimate will differ by less than one SE from the number that would have been obtained if all dwellings had been included, and about 19 chances in 20 (95%) that the difference will be less than two SEs. Another measure of the likely difference is the relative standard error (RSE), which is obtained by expressing the SE as a percentage of the estimate.

2 Due to space limitations, it is impractical to print the SE of each estimate in the publication. Instead, a table of SEs is provided to enable readers to determine the SE for an estimate from the size of that estimate. The SE table is derived from a mathematical model, referred to as the 'SE model', which is created using the data collected in this survey. It should be noted that the SE model only gives an approximate value for the SE for any particular estimate, since there is some minor variation between SEs for different estimates of the same size. Table T1 shows the SEs and RSEs that should be used for estimates of persons aged 5 to 14 years who worked in the last 12 months.

CALCULATION OF STANDARD ERRORS

3 An example of the calculation and the use of SEs in relation to estimates of people is as follows. Table 1 shows that the estimated number of children in Australia aged 5 to 14 years who worked in the last 12 months was 175,100. Since this estimate is between 150,000 and 200,000, table T1 shows the SE will be between 7,800 and 9,000, and can be approximated by interpolation using the following general formula:

4 Therefore, there are about two chances in three that the value that would have been produced if all dwellings had been included in the survey will fall in the range 166,700 to 183,500, and about 19 chances in 20 that the value will fall within the range 158,300 to 191,900. This example is illustrated in the diagram below.

5 In general, the size of the SE increases as the size of the estimate increases. Conversely, the RSE decreases as the size of the estimate increases. Very small estimates are thus subject to such high RSEs that their value for most practical purposes is unreliable. In the tables in this publication, only estimates with RSEs of 25% or less are considered reliable for most purposes. Estimates with RSEs greater than 25% but less than or equal to 50% are preceded by an asterisk (e.g.*3.4) to indicate they are subject to high SEs and should be used with caution. Estimates with RSEs of greater than 50%, preceded by a double asterisk (e.g.**0.3), are considered too unreliable for general use and should only be used to aggregate with other estimates to provide derived estimates with RSEs of 25% or less.

6 The RSEs for estimates of children who did not work and total children aged 5 to 14 years are obtained by first finding the RSE of the population estimate from table T1 then multiplying the resulting number by a factor of 0.32.

7 The following is an example of the calculation of SEs where the use of a factor is required. Table 1 shows that the estimated number of boys aged 5 to 9 years that did not work was 655,100. The SE of 655,100 children that did work can be calculated from table T1 (by interpolation) as 15,500 (rounded to nearest 100). To convert this to an SE relating to boys aged 5 to 9 years that did not work, multiply this number by the factor 0.32: 15,500 x 0.32 = 5,000 (rounded to nearest 100). The RSE of this estimate of boys aged 5 to 9 years that did not work is therefore 5,000/655,100 = 0.8%.

8 In Table 1 estimates of the total number of children aged 5 to 14 years are shown by Sex, Country of birth of child, State or territory and Area of usual residence. With the exception of Country of birth of child, these total estimates have zero sampling error as they are the items used as benchmarks.

PROPORTIONS AND PERCENTAGES

9 Proportions and percentages formed from the ratio of two estimates are also subject to sampling errors. The size of the error depends on the accuracy of both the numerator and the denominator. A formula to approximate the RSE of a proportion is given below. This formula is only valid when x is a subset of y.

10 Considering the example above, of the 175,100 children aged 5 to 14 years who worked in the last 12 months, 101,000 or 57.7% were boys. The SE of 101,000 may be calculated by interpolation as 6,400. To convert this to an RSE we express the SE as a percentage of the estimate, or 6,400/101,000 = 6.3%. The SE for 175,100 was calculated previously as 8,400, which converted to an RSE is 8,400/175,100 = 4.8%. Applying the above formula, the RSE of the proportion is:

11 Therefore, the SE for the proportion of boys aged 5 to 14 years who worked in the last 12 months is 2.4 percentage points (=(57.7/100)x4.1). Therefore, there are about two chances in three that the proportion of boys aged 5 to 14 years who worked in the last 12 months is between 55.3% and 60.1% and 19 chances in 20 that the proportion is within the range 52.9% to 62.5%.

DIFFERENCES

12 Published estimates may also be used to calculate the difference between two survey estimates (of numbers or percentages). Such an estimate is subject to sampling error. The sampling error of the difference between two estimates depends on their SEs and the relationship (correlation) between them. An approximate SE of the difference between two estimates (x-y) may be calculated by the following formula:

13 While this formula will only be exact for differences between separate and uncorrelated characteristics or subpopulations, it is expected to provide a good approximation for all differences likely to be of interest in this publication.

STANDARD ERRORS

T1 Standard errors of estimates(a)

SE

RSE

Size of estimate (children)

no.

%

100

130

130

200

200

100

300

260

87

500

350

70

700

420

60

1,000

520

52

1,500

660

44

2,000

780

39

2,500

900

36

3,000

1 000

33

3,500

1 100

31

4,000

1 150

29

5,000

1 300

26

7,000

1 600

23

10,000

1 900

19

15,000

2 400

16

20,000

2 800

14

30,000

3 450

12

40,000

4 000

10

50,000

4 500

9

100,000

6 400

6

150,000

7 800

5

200,000

9 000

5

300,000

10 900

4

500,000

13 900

3

1,000,000

19 200

2

2,000,000

26 200

1

(a) Refers to children aged 5 to 14 years who worked in the last 12 months. To calculate RSEs for other estimates in this publication, refer to paragraphs 6 and 7 of the Technical Note.

T2 Levels at which estimates have relative standard errors of 25% and 50%(a)